# Why Use a Fuzzy Partition in F-Transform?

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## Abstract

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## 1. Formulation of the Problem

#### 1.1. F-Transform: A Brief Reminder

#### 1.2. The General Idea behind F-Transform Is Very Reasonable

#### 1.3. However, Why a Fuzzy Partition?

#### 1.3.1. Mathematical Comment

#### 1.3.2. Application-Related Comment

#### 1.4. It Is Desirable to Explain the Efficiency of a Fuzzy Partition Requirement

#### 1.5. What We Do in this Paper

#### 1.6. The Structure of this Paper

#### Comment

## 2. Main Idea

#### 2.1. What if We Can Make Exact Measurements of Instantaneous Values?

- we reconstruct the values $x({t}_{1})$, $x({t}_{2})$, …, with perfect accuracy (0 measurement error), while
- the values $x(t)$ corresponding to all other moments of time t are reconstructed with no accuracy at all (the only bound on measurement error is infinity).

- we know the values $x({t}_{i})$ with finite accuracy, but
- for all other moments of time t, we know nothing (i.e., the only bound of measurement error is infinity).

#### 2.2. Main Idea

#### Comment

## 3. Case of Probabilistic Uncertainty

#### 3.1. Description of the Case

- that each measurement error $\Delta {m}_{i}$ is normally distributed with 0 mean and known standard deviation $\sigma $, and
- that measurement errors $\Delta {m}_{i}$ and $\Delta {m}_{j}$ corresponding to different measurements $i\ne j$ are independent.

#### 3.2. How Accurately Can We Estimate $X(T)$ Based on Each Measurement

#### 3.3. How Accurately Can We Estimate $X(T)$ Based on All The Measurements

#### 3.4. Discussion

- in the fuzzy partition requirement, we demand that the sum of the functions ${A}_{i}(t)$ be constant, but
- here, we have the sum of the squares.

## 4. How Uncertainties Can Be Combined in Different Approaches

#### 4.1. Towards a General Formulation of the Problem

#### 4.2. Commutativity

#### 4.3. Associativity

- we can first combine the first and the second ones, and then combine the result with the third one,
- or we can first combine the second and the third ones, and then combine the result with the first one.

#### 4.4. Monotonicity

#### 4.5. Non-Degenerate Case

#### 4.6. Scale-Invariance

#### 4.7. Discussion

**Definition**

**1.**

- for all a and b, we have $a\ast b=b\ast a$ (commutativity);
- for all a, b, and c, we have $(a\ast b)\ast c=a\ast (b\ast c)$ (associativity);
- for all a and b, we have $a\ast b\le a$ (first monotonicity requirement);
- for all a, b, ${a}^{\prime}$, and ${b}^{\prime}$, if $a\le {a}^{\prime}$ and $b\le {b}^{\prime}$, then $a\ast b\le {a}^{\prime}\ast {b}^{\prime}$ (second monotonicity requirement);
- if $a>0$ and $b>0$, then $a\ast b>0$ (non-degeneracy); and
- for all a, b, and $\lambda >0$, we have $(\lambda \xb7a)\ast (\lambda \xb7b)=\lambda \xb7(a\ast b)$ (scale-invariance).

#### Comment

**Proposition**

**1.**

#### Comment

#### 4.8. Discussion

## 5. Which Functions ${A}_{i}(T)$ Should We Choose: General Uncertainty Situation and Case of Fuzzy Uncertainty

#### 5.1. Analysis of the Problem

#### 5.2. General Conclusion

#### 5.3. Which Value $\beta $ Should We Use in the Case of Fuzzy Uncertainty

## 6. Conclusions and Future Work

#### 6.1. Conclusions

#### 6.2. Possible Directions of Future Research

- to different imprecise probability situations, and
- to situations when different functions ${A}_{i}(t)$ correspond to different types of uncertainty.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Kreinovich, V.; Kosheleva, O.; Sriboonchitta, S.
Why Use a Fuzzy Partition in F-Transform? *Axioms* **2019**, *8*, 94.
https://doi.org/10.3390/axioms8030094

**AMA Style**

Kreinovich V, Kosheleva O, Sriboonchitta S.
Why Use a Fuzzy Partition in F-Transform? *Axioms*. 2019; 8(3):94.
https://doi.org/10.3390/axioms8030094

**Chicago/Turabian Style**

Kreinovich, Vladik, Olga Kosheleva, and Songsak Sriboonchitta.
2019. "Why Use a Fuzzy Partition in F-Transform?" *Axioms* 8, no. 3: 94.
https://doi.org/10.3390/axioms8030094